Summation of series in statistical mechanics by continued exponentials

Abstract

Continued exponentials (CEs) are used to extend a series when a finite number of coefficients are known. Since the CE does not have singularities built-in (as Padé approximants and the maximum-entropy (ME) method do) it is a useful technique to estimate thermodynamic functions from partial series right up to the transition point (radius of convergence) for first-order phase transitions. The method is tested on Fisher’s cluster model (an ideal mixture of clusters with no excluded volume that contains a first-order gas–liquid phase transition with a critical point), the one-dimensional Ising model (an example of a fluid above the critical point) and the two-dimensional Ising model (an example of a fluid below the critical point where the effect of excluded volume is included). In addition, we compare the use of the CE to estimate the equation of state for hard spheres from the known partial virial series (through B 8) with the results from Padé approximants and the ME method. The results from the three methods are comparable, with the results from the CE approximation agreeing best with simulation data. CEs can also be used to approximate series that are asymptotic to a power law in the case where the exponent is known. We use the case of diffusion in one dimension as an example with power-law behavior and compare the use of CEs and the ME method to treat this system.